Abstract

Infinitesimal deformations of maximal orders over smooth alge-braic surfaces are studied. We classify which maximal orders mayadmit deformations that are not finite over their centres. The clas-sification is in terms of the algebraic surface that corresponds to thecentre of the order and the ramification divisor of the order.

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Quantizable Orders over Surfaces

Colin Ingalls

February 9, 2012

Contents

1 Introduction 2

2 Hochschild Cohomology 7

3 Grobner Bases and Deformations 13

4 The Five Term Exact Sequence 22

5 Deformations at Height One Primes 26

6 Deformations at Singularities of the Discriminant 33

7 Deformations of Orders over Surfaces 39

1 Introduction

Let X be a smooth projective surface over k, an algebraically closed fieldof characteristic zero. Let K be the field of rational functions on X, and Abe an OX-order in a central simple K-algebra. We study deformations ofA as an algebra over the base field k. We determine conditions necessaryfor the existence of an infinitesimal deformation not finite over its centre.The main result classifies which maximal orders allow such deformations.The proof has an affine case which determines the deformations of an order,and a global geometric part which classifies the possible projective surfacesand ramification divisors associated to an order that admits deformations

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not finite over its centre. The main result is the following combination ofpropositions 7.10, 7.5 and 7.16.

Theorem 1.1 Let A be a maximal order over a smooth projective surfaceX. If A admits a deformation that is not finite over its centre then X hasan effective anti-canonical divisor D, and one of the following is true

1. X is rational or birationally ruled and A ' End(V ) for some vectorbundle V ,

2. X is rational, A is ramified on D, and D is one of the following:

(a) a smooth elliptic curve,

(b) a rational curve with one node,

(c) a polygon of smooth rational curves,

3. X is birationally ruled over an elliptic curve E, A is ramified on D,and D = E0 + E ′0 where E0, E

′0 are disjoint sections of E,

4. X is a minimal K3 surface or an abelian surface and A is an Azumayaalgebra.

The affine part of the proof uses Hochschild cohomology and Grobnerbases to compute the deformations using the known local structures of theorder. Let R be a two dimensional k-algebra and let X = SpecR. LetA be a maximal order in a central simple K(R)-algebra. We will writeΘ = Derk(R,R) and Θ2 = ∧2 Derk(R,R). Let D be the curve that is thelocus of the reduced discriminant of A. We will denote the singular locus of Dby SingD. Let I be the ideal sheaf of D. There are R-modules AlgExtk(A,A)and AlgExtR(A,A) parameterizing first order deformations of A as an k-algebra and as an R-algebra respectively. Let S(A) denote the image of thenatural map AlgExtR(A,A)→ AlgExtk(A,A). The next result follows fromtheorem 5.11.

Theorem 1.2 Let R be a smooth k-algebra of dimension two. Let A be amaximal R-order. Let I be the ideal of the reduced discriminant of A. Thereis a canonical exact sequence

0→ S(A)→ AlgExtk(A,A)→ IΘ2,

and S(A) is a finite length R-module whose support contained in SingD.

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The deformations corresponding to the elements of S(A) are deformations ofA as an R-algebra, so they are finite over their centres.

Example 1.3 Let R = k[u, v], so X = SpecR is the affine plane. Let A bethe R-algebra generated by elements x, y satisfying the relations x2 = u, y2 =v, xy + yx = 0. Then A is a maximal order in the central simple algebraA⊗ k(u, v). The discriminant locus of A is the union of the coordinate axesuv = 0. The order A is generated as a k-algebra by x, y satisfying the relationyx+ xy = 0. Let k1 = k[ε]/ε2 and let A1 be the k1-algebra generated by x, ysatisfying the relation xy + yx − εxy = 0. The centre of A1 is contained ink + εA1, and so A1 is not finite over its centre.

Proposition 1.4 Let A1 be a deformation of A over k[ε]/ε2. Then there areunique α in k and f in R so that A1 is isomorphic to the k-algebra generatedby elements x, y satisfying the relation xy + yx− εα− εfxy.

This proposition follows from 6.1 and 5.11. The term α is an element ofS(A) ' k and is torsion supported at SingD. The term f is a commutator.If we lift u, v in R to u, v then [u, v] = ε4fuv. The commutator vanishes onthe discriminant locus D. The deformation A1 is finite over its centre if andonly if f = 0.

There is an exact sequence that helps in computing the deformations ofA.

Proposition 1.5 Let A be a maximal order over a smooth k-algebra R ofdimension two. There is a canonical exact sequence

0→ OutDerk(A,A)→ Θφ1→ AlgExtR(A,A)→ AlgExtk(A,A)

φ2→ Θ2

In the above sequence OutDerk(A,A) is the R-module of outer derivationson A. The proof of this proposition uses a change of base spectral sequencefor Hochschild cohomology, 2.5, and the fact that OutDerR(A,A) = 0 inproposition 4.6.

An analysis of this sequence is used to compute AlgExtk(A,A). LetNX/D = HomR(I, R/I), the normal sheaf of D in X. The natural mapΘ→ NX/D factors through AlgExtR(A,A). This gives the following six termexact sequence of kernels and cokernels.

0→ OutDerk(A,A)→ Θ(− logD)θ→ kerψ → S(A)→ T1 → cokψ → 0

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In the above sequence T1 is the module of deformations of the curve D andΘ(− logD) is module of derivations preserving I.

Local computations using Grobner bases show that θ = 0 in codimensionone and that the final four terms of the sequence are all torsion supportedin SingD. Using the result of these computations it is determined thatIm(φ2) ⊆ ∧2Θ(− logD) ' I ∧2 Θ. This completes the algebraic part of theproof.

The structure of the affine deformations is analyzed in three cases. Thesecases correspond to the possible etale local structures of the order. The nextproposition follows from [Reiner] and [Artin82a], [Artin86]. The argumentsof proposition 3.4 and theorem 3.8 (a) of [Artin82a] apply without change toprove part 2.

Proposition 1.6 Let A be an maximal order on a smooth surface X ramifiedon the discriminant curve D. Then

1. if p is not in D then A is etale locally isomorphic to a matrix algebraat p.

2. if p is a smooth point of D then A is etale locally isomorphic to astandard order at p.

3. if p is a singular point of D and A has global dimension two then thereis a list of possible complete local structures of A at p in [Artin86].

Standard orders are defined in 4.1. The deformations in the first case aredescribed in 5.1. The deformations in second case are analyzed in section5. These deformations together with reflexivity are used to prove the mainalgebraic theorem 5.11. The third case is analyzed in section 6. In this casea stronger version of the algebraic theorem is shown.

It follows from 1.2 that if an maximal order A over a projective surfaceX has a deformation not finite over its centre then H0(X,OX(−D−K)) 6= 0.Hence H0(X,OX(−K)) 6= 0. This leads to the geometric part of the proof.The possible surfaces X and ramification curves D so that H0(X,OX(−D−K)) 6= 0 are described. This is done using the classification of surfaces andthe Brauer group. Since −K is effective, X must be rational, birationallyruled, minimal K3 or abelian by [BPV]. The possible effective anti-canonicaldivisors in a rational or birationally ruled surface are listed. Exact sequencesinvolving the Brauer group of a surface limit the possible ramification curves.

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There is an exact sequence in [AM] calculating the Brauer group of asimply connected surface. It follows from the proof in [AM] that this sequenceis a complex for an aribitary projective surface. The complex restricts thepossible rational subcurves of D, and allows one to conclude the followingfact about the ramification divisor D of a maximal order 7.12. If there is anon-trivial map f : P1 → D then f−1(SingD) contains at least two points.This restricts the possible ramification curves in a rational surface to theones listed in theorem 1.1. In the case of a birationally ruled surface, thisallows us to reduce to the possibility that D is a smooth subcurve of −K.The possible ramification curves are reduced further by an exact sequenceinvolving the Brauer group of U = X −D, where D is a smooth curve in X.The following is proposition 7.17.

Proposition 1.7 Let D be a smooth curve in a smooth projective surface X,then there is a canonical exact sequence

0→ Br(X)→ Br(U)→ H1(D,Q/Z)γ→ H3(X,µ)→ H3(U, µ)

where the map γ is Poincare dual to the restriction map H1(X,µ) →H1(D,µ).

The map γ is an isomorphism if D is a section of birationally ruled surfaceX. Using this fact we show that the possible ramification divisors containedin −K is as in the statement of the theorem 1.1.

The paper is organized as follows. Section 2 recalls some facts aboutHochschild cohomology. Section 3 shows how to compute deformations andHochschild cohomology using Grobner bases 3.12. The method is basedon the method in [Anick]. The last part of section 3 applies this methodto compute the first order deformations of a certain class of orders whichincludes standard orders. In section 4 it is proved that OutDerR(A,A) =0. This fact is used to refine the 5-term exact sequence of the change ofbase spectral sequence of Hochschild cohomology. Section 5 computes thedeformations at codimension one primes and uses reflexivity to obtain themain algebraic result. Section 6 computes the module S(A) for most orders ofglobal dimension two and proves a stronger algebraic result for these orders.Section 7 generalizes the algebraic results to the global case and computesthe possible surfaces and ramification divisors of maximal orders that mayallow deformations not finite over their centres.

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Acknowledgments This paper is based on the author’s Ph.D. thesis.I would like to thank my advisor Michael Artin for many suggestions andinvaluable help.

2 Hochschild Cohomology

In this section some known facts about Hochschild cohomology are reviewed.Most of the following is in [Wb] and [Grst]. The possibly new results are thechange of base spectral sequence for Hochschild cohomology 2.5, and theresulting invariance under etale base change 2.15.

Let R be a commutative ring and let A be an R-algebra. By this it ismeant that there is a map from R to the centre of A. We say that A is aprojective R-algebra if A is projective as an R-module.

Definition 2.1 We define R1 to be a nilpotent extension of R if there is anilpotent ideal N of R1 so that R1/N = R. Suppose A is a flat R-algebra.Given a nilpotent extension R1 of R, a deformation of A over R1 is a flatR1-algebra A1 with a given isomorphism A1 ⊗R1 R ' A. Two deformationsA1 and A′1 are isomorphic if there is an R1-algebra isomorphism A1 → A′1that induces the identity map on A modulo N .

Let R be a commutative ring and A be an R-algebra. The envelopingalgebra of A over R is defined to be AeR = A⊗R Ao. A left AeR module M isan A-bimodule with a symmetric R-action.

The Hochschild cohomology of A with coefficients in M is defined to be

H iR(A,M) = ExtiAe

R(A,M).

The low dimensional groups have useful interpretations. Define the central-izer of A in M to be

MA = m ∈M such that [A,m] = 0.

Note that AA is the centre of A. There is the natural map M → DerR(A,M)defined by m 7→ [m,−], the derivation given by the commutator with m.Define the outer R-derivations from A to M to be the cokernel of this map,giving the exact sequence

0→MA →M → DerR(A,M)→ OutDerR(A,M)→ 0.

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Let A be a projective R-algebra. Consider algebra extensions of A by M

0→M → A1 → A→ 0

with M2 = 0. Let AlgExtR(A,M) be the set of isomorphism classes of theseextensions.

The proof of the following facts is in [Wb] 9.1.1, 9.2.1, 9.3.1, and [Grst]section 3.

Proposition 2.2 Let R be a commutative algebra and let A be an R-algebra,then

H0R(A,M) ' MA,

H1R(A,M) ' OutDerR(A,M).

If A is a projective R-algebra then

H2R(A,M) ' AlgExtR(A,M)

and AlgExtR(A,A) is the set of isomorphism classes of deformations of Aover R1 = R[ε]/ε2. If we let

H3R(A,A) = ObsR(A,A)

then there is a (non-linear) map AlgExtR(A,A)→ ObsR(A,A) that gives theobstruction to extending a first order deformation to second order.

In the case that A is a projective R-algebra, the Hochschild cohomologycan be calculated by a canonical, but unfortunately large complex known asthe bar complex. Let Cn

R(A,M) = HomR(A⊗n,M) be the bar cochains. Letδ : Cn

R(A,M)→ Cn+1R (A,M) be the boundary map defined by

δf(a1 ⊗ · · · ⊗ an) = a1f(a2 ⊗ · · · ⊗ an)

−f(a1a2 ⊗ a3 ⊗ · · · ⊗ an)

+f(a1 ⊗ a2a3 ⊗ · · · ⊗ an)

+ · · ·+(−1)nf(a1 ⊗ . . .⊗ an−2 ⊗ an−1)an.

It is well known that this complex computes the Hochschild cohomology [Wb]9.1.1, [Grst] section 3. The following result is proved in [Grst] section 10.

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Proposition 2.3 The Hochschild cohomology H∗R(A,A) is a graded skewcommutative algebra over the centre of A. The multiplication is induced bythe following definition on bar cochains.

(f ^ g)(a1 ⊗ · · · ⊗ an+m) = f(a1 ⊗ · · · ⊗ an)g(an+1 ⊗ · · · ⊗ an+m). (2.4)

Proposition 2.5 Let k be a commutative ring and let R be a commutativek-algebra. Let A be a flat R-algebra and let M be an A-bimodule with asymmetric R-action. Then there is a spectral sequence

Hp+qk (A,M)⇐ Hp

R(A,Hqk(R,M)) = Epq

2 .

Proof. Given an arbitrary ring map S → T , a left S-module M and a leftT -module N , there are the standard change of base spectral sequences, [Wb]5.6.3,

Extp+qS (M,N)⇐ ExtpT (TorSq (M,T ), N)) = Epq2 , (2.6)

Extp+qS (N,M)⇐ ExtpT (N,ExtqS(T,M)) = Epq2 , (2.7)

Using the spectral sequence 2.6 for the map Rek → Aek yields

Extp+qRek

(R,M)⇐ ExtpAek(Tor

Rek

q (R,Aek),M) = Epq2

Consider extending an R-bimodule M to an A-bimodule. This operationmay be writen in two ways by considering M as left Re

k-module or as anR−R-bimodule,

M ⊗RekAek ' A⊗RM ⊗R A.

Since A is flat over R, Aek is flat over Rek. Also AeR ' R⊗Re

kAek. So the above

spectral sequence collapses to give

ExtqRek(R,M) ' ExtqAe

k(AeR,M). (2.8)

The change of base spectral sequence 2.7, applied to the map Aek → AeR,gives us

Extp+qAek

(A,M)⇐ ExtpAeR

(A,ExtqAek(AeR,M)) = Epq

2 . (2.9)

Combine 2.8 and 2.9.

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Corollary 2.10 Let k be a commutative ring, R a k-algebra. Let A be aprojective R-algebra. The edge map of the above spectral sequence

H∗k(A,A)→ H0R(A,H∗k(R,A))

is a map of graded skew commutative algebras.

Proof. The edge map is given by restriction of bar cocycles. The definitionof the cup product 2.3 is compatible with restriction. Let a be in A, and f bein Cn

k (A,A). We interpret a as being in C0k(A,A). Note that (δa)(x) = [x, a].

Using the notation of [Grst] section 10 p. 85, we define the Hochschild cochainf δa ∈ Cn

k (A,A) by

(f δa)(x1 ⊗ · · · ⊗ xn) = f([x1, a]⊗ x2 ⊗ · · · ⊗ xn)

+f(x1 ⊗ [x2, a]⊗ x3 ⊗ · · · ⊗ xn)

+f(x1 ⊗ x2 ⊗ [x3, a]⊗ · · · ⊗ xn)

+ · · ·+f(x1 ⊗ . . .⊗ xn−1 ⊗ [xn, a]).

We also define (δaf) ∈ Cnk (A,A) by

(δaf)(x1 ⊗ · · · ⊗ xn) = f(x1 ⊗ · · · ⊗ xn)a− af(x1 ⊗ · · · ⊗ xn).

Gerstenhaber [Grst] section 10 p. 85, shows that the graded commutator ofthe product defined by defines a graded lie bracket on the Hochschild coho-mology H∗k(A,A). In particular he shows that f δa− δaf is a coboundary.Let i : R → A be the algebra structure map. When we restrict to R we seethat i∗(f δa) = 0. Hence i∗(δaf) = [a, i∗f ] is a coboundary.

Lemma 2.11 Let k be a commutative ring, R a Noetherian ring. M afinitely generated R-module, and N an R − k-bimodule. Let V be a flatk-module. Then

ExtpR(M,N ⊗k V ) ' ExtpR(M,N)⊗k V.

Proof. Let F ∗ → M be a resolution of M by finite free R-modules. Thestatement is true if M is a finite free R-module. Hence the two functors

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HomR(−, N ⊗k V ) and HomR(−, N) ⊗k V give isomorphic complexes whenapplied to the resolution of M . Hence they have isomorphic cohomology.Since V is flat the cohomology of HomR(F ∗, N)⊗k V is Ext∗R(M,N)⊗k V.

In the case that R is smooth, the spectral sequence 2.5 has a simplerform. We define the module of dual q-forms to be Θq

R/k = ∧q Derk(R,R).

Proposition 2.12 Let k be a commutative Noetherian ring, and let R bea smooth commutative k-algebra. Let A be a flat R-algebra so that AeR isNoetherian and let M be an AeR-module. Then

Hp+qk (A,M)⇐ Hp

R(A,M)⊗R ΘqR/k = Epq

2

Proof. Using the fact that TorRe

kq (R,R) ' Ωq

R/k as shown in [Wb] 9.4.7 andthe spectral sequence 2.6 for the map Re

k → R,

Hp+qk (R,M) = Extp+qRe

k(R,M)

⇐ ExtpR(TorRke

q (R,R),M)

' ExtpR(ΩqR/k,M).

Since ΩqR/k is a projective R-module, the spectral sequence collapses,

Hqk(R,M) ' HomR(Ωq

R/k,M)

' ΘqR/k ⊗RM.

So using lemma 2.11,

HpR(A,Hq

k(R,M)) ' ExtpAeR

(A,ΘqR/k ⊗RM) ' Hp

R(A,M)⊗R ΘqR/k.

Lemma 2.13 Let R be a commutative ring. Let S be a flat commutativeR-algebra. Let A be a R-algebra so that AeR is Noetherian. Let M be anAeR-module, then

H iS(A⊗ S,M ⊗ S) ' H i

R(A,M)⊗R S

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Proof. We have that

(A⊗ S)eS ' (A⊗ S)⊗S (A⊗ S)o ' AeR ⊗ S.

Using flat base change for Ext, [Reiner] 2.39,

ExtiAeR⊗S

(A⊗ S,M ⊗ S) ' ExtiAeR

(A,M)⊗ S.

We will use the following statement from [Wb] 9.1.7. It follows fromcomparing the bar resolutions.

Lemma 2.14 Let R be a commutative ring. Let S be a commutative Ralgebra. Let A be a projective R-algebra. Then

H iR(A,M) ' H i

S(A⊗ S,M)

.

Proposition 2.15 Let R be a commutative Noetherian ring. Let S be anetale R-algebra. Let A be a flat R-algebra so that Aek is Noetherian. Let Mbe a AeR-module. Then

Hqk(A⊗R S,M ⊗R S) ' Hq

k(A,M)⊗R S.

Proof. The spectral sequence of proposition 2.12 collapses for R→ S etalegiving

HpR(S,M) ' Hp

S(S,M).

Hence

HpR(S,M) '

M p = 00 p > 0.

The spectral sequence 2.5 gives

Hp+qk (S,M)⇐ Hp

R(S,Hqk(R,M)).

This collapses to giveHqk(S,M) ' Hq

k(R,M). (2.16)

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Equations 2.14 and 2.16 give natural isomorphisms of functors on M . Hencethe two Grothendieck spectral sequences

Hp+qk (A⊗R S,M ⊗R S)⇐ Hp

S(A⊗R S,Hqk(S,M ⊗R S)),

Hp+qk (A,M ⊗R S)⇐ Hp

R(A,Hqk(R,M ⊗R S))

are isomorphic. Hence

Hpk(A⊗R S,M ⊗R S) ' Hp

k(A,M ⊗R S).

Now use lemma 2.11.

The above change of base equation does not hold for flat extensions. Etaleextensions are needed to collapse the spectral sequence.

So using 2.13 and 2.15 we get that 2.5 gives a spectral sequence of etalesheaves. The following statement is due to K. R. Dennis and can be foundin [Wb] 9.5.6.

Proposition 2.17 Let k be a commutative ring. Let A and B be Moritaequivalent k-algebras, by the bimodules APB, and BQA. Then Aek and Be

k areMorita equivalent by sending an A-bimodule M to Q⊗M ⊗ P , and

H ik(A,M) ' H i

k(B,Q⊗AM ⊗A P ).

The following Kunneth formula is proved in [MacLane] X.7.4.

Proposition 2.18 Let k be a field. Let A and B be k-algebras. Let Mbe an A-bimodule and N be an B-bimodule. Then there are the followingisomorphisms

Hnk (A⊗B,M ⊗N) '

⊕p+q=n

Hpk(A,M)⊗Hq

k(B,M) (2.19)

3 Grobner Bases and Deformations

Anick gives a complex computing the reduced Hochschild cohomologyExtiA(k,M) by using Grobner bases in [Anick]. Using this method we willgive a short complex for computing AlgExtR(A,M). Let R be a commuta-tive ring. We recall the basic notions and definitions of Grobner bases. Let

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〈X〉 denote the free monoid generated by the well ordered set X. We willorder 〈X〉 by the degree lexicographic order that orders by degree first andthen uses lexicographic order in each degree. For m,n in 〈X〉 we say m|n ifn = pmq for some p, q ∈ 〈X〉. Suppose we have generators rii∈Λ for someideal I of R〈X〉. Suppose the leading term mi of each ri has coefficient one.We write ri = mi − ai with mi ∈ 〈X〉 and all monomials in ai smaller thanmi. We will interpret these relations as instructions for reducing elementsmodulo I. For some f ∈ R〈X〉 we say f is reducible by ri if some mi

divides some term of f . So for a reducible f , some term of f is of the formpmiq. Replacing this term by paiq and obtaining f ′ = f − priq we get areduction of f by ri. By continuing this process, since 〈X〉 is well ordered,we eventually get f , a reduction of f that is not reducible by ri. Thenotion of Grobner bases addresses the question of when this process leads toa unique reduction.

Definition 3.1 An order ideal of monomials is a subset M of 〈X〉 so thatif m is in M and n|m then n is in M . Let R be a commutative ring. Let Abe an R-algebra that is free as an R-module. Let X be a well ordered set ofR-algebra generators of A. Let I be the ideal of relations in R〈X〉 giving A.Suppose we are given a set of generators rii∈Λ of I so that mi, the leadingterm of ri, has coefficient one. Then ri is a Grobner basis if the order idealof monomials M = n ∈ 〈X〉 such that mi - n for all i is a basis of A as anR-module. We say ri is a reduced Grobner basis if in addition each ri isnot reducible by any of the rj, for j 6= i.

Given a set of generators ri of an ideal I there is an effective methodto check that these relations form a Grobner basis. Write ri = mi− ai wheremi is the leading monomial of ri. We must suppose that mi has leadingcoefficient one. If we can write miz = xmj = xyz for some x, y, z ∈ 〈X〉 withy 6= 1, we call xyz an overlap of mi and mj. We let V (2) be the set of theseoverlaps following the notation of [Anick]. These overlaps have potentiallynon-unique reductions since we may reduce mi or mj first. Consider reducingthe difference between these two choices

0 = xyz − xyz = miz − xmj ≡ aiz − xai, modulo I.

We either get zero or a new relation in I. If the reduction is zero we say theoverlaps are consistent. The next proposition is one of the main reasons thatGrobner bases are a powerful tool for computation.

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Proposition 3.2 (Bergman’s Diamond Lemma) Suppose we are given a pre-sentation of an R-algebra

A = R〈X〉/(ri)i∈Λ

with ri = mi−ai has leading coefficient one. The generators ri of the idealI are a Grobner basis if and only if the overlaps miz = xmj = xyz in V (2)

are consistent.

The proof is in [Brg] 1.2. Note that the usual algorithm for generating aGrobner basis resulting from this proposition does not always provide rela-tions with leading coefficient one. So unless R is a field, we cannot generatea Grobner basis from an arbitrary generating set X with a term order.

The following lemma is not used elsewhere, but shows what algebraicconditions are equivalent to the existence of a Grobner basis over an arbitrarycommutative ring R.

Lemma 3.3 An R-algebra A has a Grobner basis if and only if A is free asan R-module.

Proof. One direction follows immediately from the definition, so let A bean R-algebra, free as an R-module. Let 1 ∪ X = 1, xi be a basis of Aas an R-module. Consider the relations xixj −

∑ckijxk where ckij are the

structure constants or multiplication table defining the algebra. The checkof the consistency of the overlaps xixjxk exactly gives the conditions on theckij so that the algebra A is associative. So these relations give us a Grobnerbasis. Since we are choosing a very large set of generators, the Grobner basis inthe above proof is usually very inefficient for calculation. If it is used forcalculating AlgExtR(A,A) using the method below, it will provide a complexisomorphic to that of the usual bar complex.

Let A be an R-algebra presented by a reduced Grobner basis

A = R〈X〉/(mi − ai)i∈Λ. (3.4)

Let M be the resulting order ideal of monomials that forms a basis of A. LetR1 be a nilpotent extension of R by N as in definition 3.1. We will describedeformations of A by lifting the Grobner basis.

Let A1 be a deformation of A over R1. Recall that A1 is a flat R1-algebraso that A1 ⊗R1 R ' A. Let X be a set of lifts of the elements of X to A1.

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We lift all the monomials in 〈X〉 in the corresponding way. This provideslifts mi and M of mi and M and of the monomials in the ai. We lift thecoefficients of the ai arbitrarily to get ai in R1〈X〉. With these lifts we cantry to form a presentation of A1. The next lemma shows that we can presentany deformation by using these lifts and making infinitesimal changes of therelations on A. The infinitesimal changes will have coefficients in N andmonomials in the order ideal M .

Lemma 3.5 Let A1 be a deformation of A over R1, where A is presented by3.4, then

A1 ' R1〈X〉/(mi − ai − αi), (3.6)

where the αi are linear combinations of the monomials in M with coefficientsin N .

Proof. By the nilpotent version of Nakayama’s lemma [E] Ex.4.6, we seethat M generates A1 as an R1-module. Since A1/NA1 ' A, the relationsmi − ai are satisfied modulo N . So A1 has a presentation as above withαi in NA1. We use these relations to reduce the monomials in the αi thatare divisible by some mj. Each time we carry out such a reduction, thecoefficients of the reducible monomials are in higher powers of N . Since Nis nilpotent we eventually get reductions of the αi with monomials in M . Since all deformations can be presented in the above manner, we will fix a liftof the presentation of A. We will consider whether an infinitesimal change ofthe relations of this fixed presentation gives a deformation. If we are givenan algebra with a presentation as in 3.6, we can determine if A1 is flat overR1.

Lemma 3.7 Let A1 be presented by

A1 = R1〈X〉/(mi − ai − αi) (3.8)

where mi − ai is a reduced Grobner basis modulo N , and the αi are linearcombinations of the monomials in M with coefficients in N . Then A1 is adeformation of A over R1 if and only if mi − ai − αi is a Grobner basis.

Proof. If mi − ai − αi is a Grobner basis then A1 is a free R1-moduleby definition 3.1, and so A1 is flat. Conversely, suppose mi − ai − αi isnot a Grobner basis. Bergman’s diamond lemma 2.1 says that some overlapxyz = xmi = mjz gives a non-trivial relation f of A1. Since the overlaps

16

are consistent modulo N , the relation f has coefficients in N . This gives usnon-zero element of N ⊗R1 A1. Consider applying the functor − ⊗R1 A1 tothe exact sequence

0→ N → R1 → R→ 0.

The kernel of the map A1 → R⊗ A1 is NA1. So we have a nonzero elementin the kernel of the map N ⊗R1 A1 → NA1 showing that A1 is not flat.

We define embedded deformations, EmbDef(A,R1), to be the set of allideals of R1〈X〉 that satisfy the conditions of the above proposition. SoEmbDef(A,R1) contains a representative of every deformation. In the casethat N2 = 0 the overlap checks give R-linear conditions on the αi. We nowmake the assumption that R1 is an R-algebra and N2 = 0. Let V = mi,the set of leading monomials. Let RV be the free R-module generated byV . We write αi = α(mi) for some α ∈ HomR(RV,N ⊗A). These will be ourinfinitesimal changes in the relations. By lemma 3.5 we may consider theαi to give values in N ⊗ A. Instead of lifting the ai randomly as before, wemay use the algebra structure map R→ R1 to lift the ai. We thus obtain acanonical trivial deformation of A. We write V (2) for the set of overlaps.

Corollary 3.9 Let R1 be a nilpotent extension of R by N so that R1 is anR-algebra and N2 = 0. Then there is a map

φ : HomR(RV,N ⊗ A)→HomR(RV (2), N ⊗ A).

with kernel isomorphic to EmbDef(A,R1). The deformation correspondingto α in kerφ may be presented by

A1 = R1〈X〉/(mi − ai − α(mi)).

Proof. All flat deformations may be written in the above form by proposition3.5. Such an algebra A1 is a deformation if and only if the relations form aGrobner basis by lemma 3.7. Each consistent overlap on A gives a syzygyon the relations. Let xmi = xyz = mjz be an overlap. We have that

0 = xyz − xyz ≡ xai − ajz, modulo I.

Since A is presented by a Grobner basis the overlap computation yields anexpression in R〈X〉∑

gkl(mk − ak)hkl = xai − ajz= xyz − xyz + xai − ajz= −x(mi − ai) + (mj − ai)z.

17

Now we compute the overlaps of the algebra A1 presented by 3.8. Theoverlaps are satisfied modulo N so we obtain the following on computing theoverlaps

xαi − αjz +∑

gklαkhkl.

This expression must be reducible to zero by the Grobner basis of A in orderfor A1 to be flat. Hence we require that

xαi − αjz +∑

gklαkhkl ≡ 0, modulo I. (3.10)

We assemble these conditions into an R−linear map φ by sending the overlapxyz in V (2) to the above equation 3.10.

We wish to identify isomorphic deformations so we can calculateAlgExtR(A,A).

Lemma 3.11 Let A1, A′1 be deformations of A. Let ψ be an R1-isomorphism

A1 → A′1 so that ψ induces the identity on A. Then ψ may be lifted to anautomorphism Ψ of the deformation R1〈X〉 of R〈X〉 so that

R1〈X〉 A1

R〈X〉 A

R1〈X〉 A1

Ψ ψ

commutes.

Proof. Since R1〈X〉 is free we may lift ψ to Ψ so that the diagram commutes.Since Ψ is the identity modulo the nilpotent ideal N , Ψ is an isomorphism.Let Aut(R1〈X〉;R〈X〉) denote the group of automorphisms of the deforma-tion R1〈X〉. This group acts on EmbDef(A,R1). The quotient of this actiongives the set of isomorphism classes of deformations of A over R1.

Since N2 = 0, Aut(R1〈X〉;R〈X〉) ' DerR(R〈X〉, N ⊗ R〈X〉). So wemay act on the kernel of the map φ by this group. A derivation δ ∈DerR(R〈X〉, N ⊗R〈X〉) acts additively on α ∈ HomR(RV,N ⊗ A) by

(δ + α)(mi) = δ(ri) + α(mi).

18

So only the residue of δ in N ⊗ A affects α. So the group

DerR(R〈X〉, N ⊗ A) ' HomR(RX,N ⊗ A)

acts additively on kerφ.

Proposition 3.12 The cohomology of the middle term of the complex

0→ HomR(RX,N ⊗ A)→ HomR(RV,N ⊗ A)→ HomR(RV (2), N ⊗ A)

is AlgExtR(A,A⊗N) and classifies deformations of A over R1. The kernelof the first map is DerR(A,A⊗N).

Proof. The first statement follows from the above discussion. A derivationδ ∈ DerR(A,N ⊗A) may be specified by its action on the generators δ(x) forx ∈ X so that it vanishes on the relations mi − ai. This is exactly the thekernel of the first map of the above complex.

We will provide an application of the proceeding section. Define an R-algebra A to be diagonal if it is generated by x, y with relations xn− a, yn−b, yx − ζxy, for some a, b ∈ R, and ζ a primitive nth root of unity. LetR1 = R[ε]/ε2.

Proposition 3.13 Let R be a commutative k-algebra. Let a, b be regularelements in R.

1. Any first order deformation over R1 of the diagonal algebra A is iso-morphic to an R1-algebra A1 generated by x, y satisfying the relationsxn − a− αε, yn − b− βε, yx− ζxy − γε, where α, β, γ ∈ R.

2. If A1 is given by α, β, γ and A′1 is given by α′, β′, γ′ then A1 ' A′1 iffα ≡ α′ modulo aR, β ≡ β′ modulo bR and γ ≡ γ′ modulo aR + bR.Thus

AlgExtR(A,A) ' R

(a)⊕ R

(b)⊕ R

(a, b).

Proof. Note that the relations of the diagonal algebra A give a Grobnerbasis. The resulting order ideal of monomials M = xiyj such that i, j =0 . . . n − 1 is an R-basis of A. The overlaps are xn+1, yn+1, ynx, yxn. Usingthe complex of proposition 3.12 the cohomology of

Hom(RX,A)→ Hom(RV,A)→ Hom(RV (2), A)

19

is AlgExtR(A,A). In our present case RX,RV,RV (2) are free modules onthe formal generators

RX = Rx⊕RyRV = Rxn ⊕Ryn ⊕Ryx

RV (2) = Rxn+1 ⊕Ryn+1 ⊕Rynx⊕Ryxn.

We will let α, β, γ denote the coordinates of the dual generators of xn, yn, yxin Hom(RV,A). They form the infinitesimal changes of the correspondingrelations. We will write

α(x, y) =n−1∑i,j=0

αijxiyj

with αij in R. The four overlap computations corresponding to the elementsof V (2) yield the following conditions on α, β, γ.

[x, α] = 0, [y, β] = 0 (3.14)

[β, x]−n−1∑k=0

ζkyn−1−kγyk = 0 (3.15)

[α, y] +n−1∑k=0

ζkxkγxn−k−1 = 0. (3.16)

Observing the condition 3.14,

0 = [x, α] = xα(x, y)−α(x, y)x = xα(x, y)−xα(x, ζy) =∑

(1−ζj)αijxi+1yj,

we see that αij = 0 for j 6= 0. Hence we rewrite α(x, y) = α(x). So

α(x) =∑

αixi, with αi ∈ R,

and similarly,

β(y) =∑

βiyi, with βi ∈ R.

We now examine the effect of the last two conditions. The condition 3.16imposed by the overlap yxn gives us

α(x)y − α(ζx)y + xn−1

n−1∑k=0

ζkγ(x, ζn−k−1y) =

20

n−1∑i,j,k=0

ζk+j(n−k−1)γijxi+n−1yj +

n−1∑i=0

(1− ζ i)αixiy = 0.

Since∑

k ζk+j(n−k−1) = 0 unless j = 1 we see that the above condition

involves only linear terms in y and simplifies to

n−1∑i=0

nζ−1γi1xi+n−1y +

n−1∑i=0

(1− ζ i)αixiy = 0.

Separating this equation by the R-basis xiyj and solving,

αi =nζ−1a

(ζ i − 1)γi+1,1

for i = 1, . . . , n− 2. Also we see

αn−1 =nζ−1

(ζ−1 − 1)γ01

and γ11 = 0. We obtain similar equations relating β and γ by symmetry. Sowe conclude that α0, β0, are arbitrary, and γij are arbitrary for j 6= 1, i 6= 1,and that α1, . . . , αn−1, β1, . . . , βn−1 are determined by γi1, γ1j by the regular-ity hypothesis.

This completes the calculation of the kernel. We now need the imageof Hom(RX,A). We let d(x) =

∑dij(x)xiyj, d(y) =

∑dij(y)xiyj be the

coordinates of the generators of Hom(RX,A) dual to x, y. They representthe infinitesimal changes in x and y. We calculate that γ is taken modulo

d(y)x+ yd(x)− ζd(x)y − ζxd(y) =∑(ζj − ζ)dij(y)xi+1yj +

∑(ζ i − ζ)dij(x)xiyj+1.

We also see that α is taken modulo∑i,j,k

dij(x)ζ−j(1+k)xi+n−1yj.

The terms in the sum are zero unless j = 0, so it simplifies to∑di1(x)nxi+n−1.

21

Each αi except αn−1, α0 is in aR, from the above calculation. So we cankill each αi for i = 1, . . . , n − 1. We see that we can take α0 modulod11(x)xn = d11(x)a. Lastly, we examine how γ is taken modulo the aboveexpression. By examining the effect on the basis xiyj we see that we cankill each γij where i, j 6= 0, 1 by using dij(x) or dij(y) with i, j = 1, . . . , n−2.We can also kill γi1 by using di0(x) for i = 0, 2, 3, . . . , n− 1. This is the sameway we killed αi above for i = 1, . . . , n−1. By symmetry we get similar resultsfor β. Finally we see γ00 can be taken modulo d0,n−1(x)yn and dn−1,0(y)xn.

4 The Five Term Exact Sequence

Let k be a characteristic zero algebraically closed field. Let Z be a normaltwo dimensional k-algebra with field of fractions K. Let A be a maximalZ-order in a central simple K-algebra. Let R be a smooth two dimensionalsubalgebra of Z so that Z is a finite projective R-module. In this section wewill show that OutDerR(A,A) = 0 and use this fact to get a refined five termexact sequence from the spectral sequence 2.5.

Definition 4.1 Given a commutative domain R and ideal p we define thestandard R-order in K(R)n×n to be

Stann(R, p) =

R R R . . . Rp R R . . . Rp p R . . . R...

.... . .

...p p p . . . R

The following proposition follows from [Reiner] 39.14 and [Janusz] theorem4. Let ζ be a primitive nth root of unity. Suppose p is the principal ideal ofR generated by some t in R. In this case the standard order Stann(R, tR)may be presented as the R-algebra generated by x, y satisfying the relationsxn − t, yn − 1, yx− ζxy.

Proposition 4.2 Let k be a field of characteristic zero. Let Z be a normaldomain over k. Let p be a height one prime of Z. Let A be a maximal Z-order that is ramified at p. There is a finite etale extension Z ′ of the d.v.r.

22

Zp so that A′ = A⊗ Z ′ is Morita equivalent to a standard order

A′ ' (Stann(Z ′, pZ ′))m×m

We will show that OutDerR(A,A) = 0 and we will calculate AlgExtR(A,A)explicitly. We first need some facts about derivations and we recall somebasic facts about reflexive modules.

Lemma 4.3 A Z-module is reflexive if and only if it is projective as anR-module.

Proof. A reflexive Z-module M has depth two since the dimension of Z istwo. So M has depth two as an R-module. Since R is smooth of dimensiontwo, the Auslander-Buchsbaum formula [E] 19.9 shows that M is a projectiveR-module. Conversely, if M is projective as an R-module then its depth asa Z-module is two and hence it is reflexive.

Lemma 4.4 There is a canonical exact sequence

0→ DerZ(A,A)→ DerR(A,A)→ DerR(Z,Z).

Proof. The only non-trivial fact needed is that if δ ∈ DerR(A,A) thenδ(Z) ⊆ Z. Let a ∈ A, z ∈ Z. We have

[a, δz] = [δa, z] + [a, δz] = δ([a, z]) = 0.

Hence a derivation of A takes central elements to central elements. Let rtr : A→ Z be the Z-linear map reduced trace as defined in section 9 of[Reiner]. We define sA to be the set of traceless elements of A.

Lemma 4.5 The modules A,DerR(Z,Z),DerR(A,A),DerZ(A,A) and sAare reflexive Z-modules.

Proof. It is shown in [AG] 1.5 that A is a reflexive Z-module. In [EG] 3.7it is shown that the kernel of a map of reflexive modules on a normal surfaceis reflexive. So proposition 3.12 shows that DerR(Z,Z) and DerR(A,A) areboth reflexive. Since sA is defined to be the kernel of the reduced trace, itis reflexive. Finally the sequence of lemma 4.4 shows that DerZ(A,A) isreflexive.

23

Proposition 4.6 Let k be a field of characteristic zero. Let Z be a normaldomain of dimension two over k with field of fractions K. Let A be a maximalZ-order. Then OutDerZ(A,A) = 0.

Proof. We localize at a height one prime p of Z. Using proposition 4.2we extend coefficients to Z ′ etale over Z so that A′ is Morita equivalent toa standard order. Using the Morita invariance of Hochschild cohomology ofproposition 2.17 and lemma 4.7, OutDerZ′(A

′, A′) = 0. By the etale basechange equation 2.14, we have OutDerZ(A,A)⊗Z ′ = 0. Since is Z ′ is finite,it is faithfully flat over Zp. So we see that OutDerZ(A,A)⊗ Zp = 0.

So the support of OutDerZ(A,A) has dimension at most zero. The re-flexive module sA is isomorphic to A/Z. So OutDerZ(A,A) is a cokernel ofa map of reflexive modules

0→ sA→ DerZ(A,A)→ OutDerZ(A,A)→ 0.

Let m be an associated maximal prime ideal of OutDerZ(A,A) with residuefield κ. Applying the functor ExtiZ(κ,−) to the above exact sequence we seethat HomZ(κ,OutDerZ(A,A)) = 0. So OutDerZ(A,A) = 0.

Lemma 4.7 Let Z be a commutative domain with field of fractions K. Letpij|i, j = 1 . . . n be a collection of non-zero ideals of Z, so that pkk = p1k =Z for all k and pijpjk ⊆ pik. Let A be the order obtained by assembling thepij into a matrix in the natural way. Then OutDerZ(A,A) = 0.

Proof. Let δ ∈ DerZ(A,A). Then δ extends to a derivation in DerK(A ⊗K,A ⊗ K) ' DerK(Kn×n, Kn×n). Since all derivations on Kn×n are innerby [FD] theorem 3.22, we represent δ by the commutator with a matrixd = (dij) in Kn×n. We may suppose that d11 = 0 by subtracting d11I ifnecessary. Since d represents a derivation on A we have that [d,A] ⊆ A. Thematrix units ekk, e1k are in A for all k so the matrices

[d, ekk] =

0 . . . 0 d1k 0 . . . 00 . . . 0 d2k 0 . . . 0...

......

......

−dk1 −dk2 . . . 0 . . . . . . −dkn0 . . . 0

... 0 . . . 00 . . . 0 dnk 0 . . . 0

24

and

[d, e1k] =

−dk1 −dk2 . . . −dkk . . . −dk,n−1 −dkn

0 . . . 0 d21 0 . . . 00 . . . 0 d31 0 . . . 0...

......

... 00 . . . 0 dk1 0 . . . 0

,

are in A. So each dij ∈ pij and so d ∈ A.

Lemma 4.8 Let k be a field of characteristic zero. Let Z be a normal domainof dimension two with field of fractions K. Let R be a smooth subring of Zso that Z is a finite projective R module. Let A be a maximal Z-order. ThenOutDerR(A,A) = 0.

Proof. Since K is a finite separable field extension of the field of fractionsK(R), of R, we have that DerK(R)(K,K) = 0. So the projective R-moduleDerR(Z,Z) is zero at the generic point and so must be zero. So by lemma4.4, DerZ(A,A) ' DerR(A,A). Hence OutDerZ(A,A) ' OutDerR(A,A). Byproposition 4.6 we have OutDerZ(A,A) = 0.

Let Θ = Derk(R,R), the module of vector fields on SpecR. Let Θ2 =∧2Θ. The 5-term exact sequence from the spectral sequence 2.12 is

0→ OutDerR(A,A)→ OutDerk(A,A)→

ΘR ⊗ Z → AlgExtR(A,A)→ AlgExtk(A,A)

.In the case that OutDerR(A,A) = 0, the E1,−

2 column of the spectralsequence of proposition 2.12 is zero. So we get the following 5-term exactsequence using lemma 4.8.

Proposition 4.9 There is a canonical exact sequence

0 → OutDerk(A,A)φ1→ ΘR ⊗ Z

p→ AlgExtR(A,A)q→ AlgExtk(A,A)

φ2→ Θ2R ⊗ Z

(4.10)

The maps of the above sequence all have useful interpretations. Themap p gives a deformation of A by applying an infinitesimal automorphism.

25

So the multiplication changes by a derivation. Let δ be a derivation inDerk(R,Z) ' ΘR⊗Z. Let R1 = R⊕εZ be the trivial algebra extension of Rby Z. Let A1 be the tensor product defined by the pushout of the followingdiagram

A[ε]/ε2 −−−→ A1x xR1

1+εδ−−−→ R1.

The above algebra A1 will be a deformation of A over R corresponding tothe derivation δ.

The map φ2 also has a useful interpretation. Let A1 be a deformation ofA over k1 = k[ε]/ε2. Let u, v in A1 be lifts of u, v ∈ R. Let a be in A1. Notethat [a, u] is in εA1. So the Jacobi identity gives

[a, [u, v]] + [v, [a, u]] + [u, [v, a]] = [a, [u, v]] = 0

So [u, v] ∈ εZ and only depends on u, v. So given a deformation of A we getan element of HomR(Ω2

R, Z) ' Θ2R ⊗ Z. We define the image of this map to

be the bracket of the deformation. A1 is finite over its centre if and only ifits bracket vanishes.

The following statement is a corollary of proposition 2.10

Corollary 4.11 The image of the map φ2 : AlgExtk(A,A) → Θ2R ⊗ Z con-

tains ∧2 Im(φ1).

5 Deformations at Height One Primes

We will now analyze the deformations of an order in codimension one and usereflexivity to obtain the main algebraic theorem 5.11. Using the sequence4.10 we see that we need to compute AlgExtR(A,A) to find AlgExtk(A,A).

We first analyze the deformations of an order at the generic points of thesurface. At the unramified points, the order A is etale locally isomorphicto a matrix algebra over its centre. The deformations of such an algebraare described by the following proposition. Since A is separable at thesepoints, the spectral sequence 2.12 collapses to give that AlgExtR(A,A) 'AlgExtk(R,R), but the actual correspondence of the deformations followseasily from facts about lifting idempotents.

26

Proposition 5.1 Let k be a commutative ring. Let R be a flat k-algebra. LetA = Rn×n, and let k1 be a nilpotent extension of k. Let A1 be a deformationof A over k. Then A1 ' Rn×n

1 where R1 is a deformation of R over k1.

Proof. Since matrix units lift modulo a nilpotent ideal, [Rowen]1.1.25,1.1.28, we lift the matrix units of A to A1. The matrix units givean isomorphism A1 ' Rn×n

1 for some R1. Since A1 is flat over k1, so is R1.We also have R1 ⊗k1 k ' R.

We now analyze the deformations along smooth points of the discriminantcurve D. If R is a d.v.r. with parameter t then Sanada shows that theHochschild cohomology of A = Stann(R, tR) is periodic with H2i

R (A,A) =R/tR and H2i−1

R (A,A) = 0 for i > 0, [Sanada] Proposition 1. The nextresult overlaps with this statement. Let R1 = R[ε]/ε2. Using propositions3.13 and 2.17, we get the following result.

Corollary 5.2 If A = Stann(R, tR)m×m then

1. AlgExtR(A,A) ' R/tR.

2. a deformation A1 over R1 is Morita equivalent to an algebra generatedover R1 by by x, y satisfying the relations xn− (t+εt′), yn−1, yx− ζxyfor some t′ ∈ R.

3. A1 ' Stann(R1, (t+ εt′)R1)m×m.

The next corollary gives an application of the above exact sequence 4.10.

Corollary 5.3 Let R = k[u] and let B = Stann(R, uR). Let S = k[u, v] andlet A = Stann(S, uS). Then

Z(B) = R Z(A) = S

OutDerk(B,B) ' uΘR OutDerk(A,A) ' uS∂

∂u⊕ S ∂

∂vAlgExtk(B,B) = 0 AlgExtk(A,A) ' uΘ2

S

Proof. It is easy to compute the centre of B and OutDerk(B,B) directly.We let R = Z and we see that AlgExtR(B,B) ' k by 5.2. We also have

27

that ΘR = R ∂∂u, and Θ2

R = 0. Using the exact sequence 4.10 we get

0→ uR∂

∂u→ R

∂

∂u→ k → AlgExtk(B,B)→ 0

Hence we conclude that AlgExtk(B,B) = 0. Since A ' B ⊗k k[v], theKunneth formula for Hochschild cohomology 2.18 computes the answers forA. Let R be a regular local ring of dimension two with field of fractions K. LetA be a maximal R-order in a central simple K-algebra of index n. Let A1 bea deformation of A over R1. Since A is reflexive, A is a free R-module. Letxi|i = 0 . . . n2− 1 be a basis of A. Let rtr : A→ R be the reduced trace ofA as in [Reiner] section 9. Recall that the discriminant of A is

∆ = det(rtr(xixj)).

We factor ∆ = f e11 · · · f enn into irreducible factors. Let d be the reduceddiscriminant d = f1 · · · fn. The reduced discriminant is defined up to multi-plication by a unit. The algebra A[d−1] is an Azumaya R[d−1]-algebra.

Lemma 5.4 Let A be an Azumaya algebra. Let A1 be a deformation of Aover R1. Then A1 is the trivial deformation and A1 is Azumaya over R1.

Proof. Since A is Azumaya, A is a separable AeR-module. SoAlgExtR(A,A) = 0. So A1 must be the trivial deformation A1 ' A ⊗R R1.

Lemma 5.5 Let A1 be a deformation of the maximal order A over R1. Leta1 ∈ A1. Let a ∈ A be the residue of a1 modulo εA. Then there is a reducedcharacteristic polynomial of a1 in R1[t], extending the reduced characteristicpolynomial of a in R[t].

Proof. Since A[d−1] is Azumaya, so is A1⊗R[d−1]. Hence there is a reducedcharacteristic polynomial rch : A1 ⊗R R[d−1]→ R1[d−1, t], [KnOj] IV.2. Leta1 be in A1 with characteristic polynomial f(t). Now when we localize at aheight one prime p of R, there is a finite etale extension R′ of Rp so that A1⊗R′ is of the form in corollary 5.2. Since these orders are subrings of matrixrings, their reduced characteristic polynomials certainly have coefficients in

28

R′. So we see that the coefficients of f(t) are in R′1 ∩ R1[d−1]. Since R1 is afinite etale extension R′1 ∩R1[d−1] = Rp[ε]/ε

2. Since R is normal,

R =⋂

ht p=1

Rp.

Hence the coefficients of f(t) are in R1.

Given an ideal I of R the set of deformations of I in R correspond toelements of NI/R = HomR(I, R/I), the module of normal vector fields toSpecR/I in SpecR. A deformation of A in AlgExtR(A,A) gives a deforma-tion of the discriminant ideal ∆R in R.

Lemma 5.6 There is an canonical R-module map

AlgExtR(A,A)→ HomR(∆R,R/∆R).

Proof. This map is given by the reduced trace. Let xi be an R1-basis ofA1. Then the deformation of the discriminant ideal is given by

∆ 7→ 1

ε(det(rtr(xixj))−∆).

Let q = ∆/d = f e1−1

1 · · · f en−1n . There is a natural injective map

Hom(dR,R/dR)→Hom(∆R,R/∆R).

This map is the composition of injective maps

Hom(dR,R/dR)s→ Hom(∆R, qR/∆R)→ Hom(∆R,R/∆R).

The map s is given by tensoring with qR.We letD = V (d), the discriminant locus inX = SpecR. LetNX/D denote

the module of normal vector fields HomR(dR,R/dR) where X = SpecR andD = V (d). A deformation of A in AlgExtR(A,A) also gives a deformation ofthe reduced discriminant ideal dR in R.

Proposition 5.7 There is a canonical R-module map

ψ : AlgExtR(A,A)→ NX/D.

The natural map ΘX → NX/D factors through this map.

29

Proof.It suffices to show that the map of lemma 5.6 takes ∆ into qR/∆R. Using

4.2 and 5.2 we can verify this statement for R′ etale over Rp for any heightone prime p of R. Hence it follows for R. The last statement follows fromthe description of the maps.

Lemma 5.8 Let R be a d.v.r with parameter t. Let A = Stann(R, tR)m×m.Then the map φ : AlgExtR(A,A)→ NtR/R is an isomorphism.

Proof. The discriminant of A is ∆ = tn and the reduced discriminant isd = t. The discriminant of a deformation of A as described in 5.2 is (t +εt′)n = (tn + nεt′n−1). The map from AlgExtR(A,A) → HomR(∆R,R/∆R)sends t′+tR inR/t ' AlgExtR(A,A) to the map defined by ∆ 7→ nt′n−1+∆R.So the image of of t′ + tR in HomR(dR,R/dR) is the map d 7→ nt′ + dR.

Let Θ(− logD) be the kernel of the map ΘX → NX/D. So Θ(− logD)is the module of those vector fields on X that are tangent to D, or alter-natively derivations δ on R that take dR into dR. Let T1 be the module ofdeformations of D as described in [Sch]. It is known that T1 is the cokernelof the map ΘX → NX/D. See [Artin76] for a proof of this fact. We denote byS(A) the cokernel of the map ΘR → AlgExtR(A,A).

Lemma 5.9 If we have maps Aψ→ B

φ→ C in an abelian category then thereis a canonical exact sequence

0→ kerψ → kerφψ → kerφ→ cokψ → cokφψ → cokφ→ 0.

The proof is a diagram chase.

We use the above lemma for the maps ΘX → AlgExtR(A,A)ψ→ NX/D.

This gives the following diagram, with zeroes on the periphery. The sequence

30

wrapping around the triangle in the centre is exact.

Θ(− logD) kerψ

OutDerk(A,A) ΘX AlgExtR(A,A) S(A)

0 NX/D

cokψ T1.

θ

ψ

(5.10)

Theorem 5.11 Let A be a maximal R-order with discriminant locus D.Then

1. The module S(A) is a finite length R-module.

2. There is an exact sequence 0→ S(A)→ AlgExtk(A,A)→ IDΘ2X .

3. There is a natural map S(A)→ T1.

4. If θ = 0 then

(a) OutDerk(A,A) ' Θ(− logD).

(b) AlgExtk(A,A) ' S(A)⊕ IDΘ2X .

Proof. The map ψ is an isomorphism and so the map θ is zero for standardorders by 5.8, so the first statement follows. The statments 3 and 4a follow

31

immediately from the diagram 5.10. To show 2 we will consider the mapφ2 : AlgExtk(A,A) → Θ2

X from the exact sequence 4.10. We must showthat the image of φ2 is contained in IDΘ2

X . There is a natural isomorphism∧2Θ(− logD) ' IDΘ2

X . Recall that the cup product of Hochschild cohomol-ogy for H∗k(A,A) is R-linear 2.3 and that the edge map of the spectralsequence gives us a map of graded skew commutative algebras 2.10. Hencewe get the diagram

∧2 OutDerk(A,A)∧2φ1−−−→ ∧2Θ

^

y ∥∥∥ ^

AlgExtk(A,A) −−−→φ2

Θ2X

(5.12)

where the horizontal maps are given by the cup product. Since we have anisomorphism on the right, and the map φ1 factors through Θ(− logD) we getthe following commutative square

∧2 OutDerk(A,A) −−−→ ∧2Θ(− logD)

^

y y^AlgExtk(A,A) −−−→

φ2Θ2X .

(5.13)

The left cup product map is an isomorphism when we localize at a codimen-sion one prime by 5.3 and 5.2, and since ∧2Θ(− logD) ' IDΘ2

X , the module∧2Θ(− logD) is reflexive. So we use the following lemma 5.14 to concludestatment 2. The statement 4b follows from 4.11 and the fact that IDΘ2

X

is torsion free.

Lemma 5.14 Suppose we have the following commutaive diagram of R-modules

Pb−−−→ Q

a

y ydT −−−→

cS.

Suppose that Q is reflexive and that the map a becomes an isomorphismupon localizing at height one primes. Then there is a map e : T → Q so thatde = c.

32

Proof.Let ∗∗ denote the reflexive hull, and let iM : M → M∗∗ be the natural

map from M to its reflexive hull. Upon taking reflexive hulls, the map a∗∗

becomes an isomorphism, and so the desired map is i−1Q b∗∗a∗∗−1iT as in the

following diagram.

P Q

P ∗∗ Q∗∗

T S

T ∗∗ S∗∗

b

a

iQ

a∗∗

b∗∗

iT

6 Deformations at Singularities of the Dis-

criminant

For this section we suppose that A has global dimension two. In this casewe can prove that the map θ of the diagram 5.10 is zero and so get thestronger statement of 5.11. We will compute the diagram 5.10 for all ofthe possible complete local structures except for types IV’ and V’. It wasverified that θ = 0 for these algebras by using Macaulay 2. The Macaulaycomputation used the same techniques as the other types, but used the basisof A as an R-module as algebra generators. Since the number of generatorsand relations was much larger it was not possible to do the computation byhand.

The possible maximal orders A of global dimension two over the com-plete local ring R = k[[u, v]] are classified in [Artin86] 3.4, 5.31-5.33. Wesummarize the main result of that paper in the following table.

33

Type Order A Reduceddiscriminant d

Ik xk − u uvyk − vyx− ζkxy

II ik x2 − u2i+1 v(v − uk+1)y2 − uk−2ivyx+ xy − 2v

IIIk x2 − v u(v2 − u2k+1)y2 − uvyx+ xy − 2uk+1

IV x3 − v v(v − u2)y3 − u2 + vyx− ζ3xy

IV ′ 1, w, w2, xw2,wxw2, ux, uxw,uwx, uwxw

V x2 − v v(v2 − u3)y2 − v2 + u3

yx+ xyV ′ 1, ux, y − v,

u−1x(y − v)

In the above table k is an arbitrary integer > 1, ζk is a primitive kth root ofunity. For type II ik, i is chosen so that 0 6 i 6 k/2. The types IV ′ and V ′

are described by an R-basis of A. They satisfy the same relations as typesIV and V . For type IV ′, w = u−1(x+ y)2.

We will compute S(A) for all of the above algebras except IV ′ and V ′.It was verified that θ = 0 for these algebras by using Macaulay 2. The sametechniques were used, but since IV ′ and V ′ are specified by bases rather thana small number of relations, the computation is much larger. The followingtable describes the module S(A) and its relation to T1 through the mapS(A) → T1 → cokψ → 0. The module S(A) is given by its relations on theR-module generators α, β, γ which correspond to the infinitesimal changes ineach of the three relations describing A as an R-algebra. The module T1 hasa single generator and so is described by the annihilator ideal. The cokernelcokψ is described in the same manner. The map S(A)→ T1 is described in

34

the proof below.

Type dimS(A) S(A) T1 cokψIk 1 α, β, uγ, vγ u, v u, vII ik 2k + 1 γ + uk−2iβ, vα, 2v − uk+1 u2i+1, v

u2k−2i+1β, (2i+ 1)u2iα u2k+1

+(k − 2i)uk−2i−1vβ,u2i+1α, (2v − uk+1)β

IIIk 2k + 2 α + uβ, uk+2β, vβ, uv, uk+1, vuk+1γ, uvγ, v2γ v2 − (2k + 2)u2k+1

2u2k + 1β + vγIV 3 α− β, uα 2v − u2, u3 u2, v

vα, u2γ, vγV 7 α + 2vβ, u2β 3v2 − u, u2v u3, v

v2β, vγ, u3γ

We use the description of the deformations of diagonal algebras in 3.13.These calculations will apply to cases Ik, IV, and V . We now suppose a, bform a regular sequence as is true in the cases we are considering. SinceID = dR = abR is a principal ideal, we identify NX/D with R/(ab). Recallthat

AlgExtR(A,A) ' R

(a)⊕ R

(b)⊕ R

(a, b).

Computing the change in the reduced discriminant d = ab, we write the mapAlgExtR(A,A)→ NX/D as the map

R

(a)⊕ R

(b)⊕ R

(a, b)

(b,a,0)−→ R

(ab).

We get the exact sequence

0→ R

(a, b)→ AlgExtR(A,A)→ NX/D →

R

(a, b)→ 0.

We now compute the diagram 5.10. If we write ΘR = R ∂∂u⊕ R ∂

∂v, the map

ΘR → AlgExtR(A,A) may be written asau avbu bv0 0

.

35

This map restricted to Θ(− logD) is zero since its image must lie in thekernel of AlgExtR(A,A)→ NX/D.

We immediately get

Corollary 6.1 Let A be a diagonal algebra over R so that a, b form a regularsequence. Then OutDerk(A,A) ' Θ(− logD) and

0→ R

(a, b)→ S(A)→ T1 →

R

(a, b)→ 0

is exact.

It is now an easy matter to obtain S(A) for diagonal algebras as in the table 6.We now compute the deformations for the cases II ik and IIIk. We first

compute AlgExtR(A,A) for a more general algebra including these two cases.We define an R-algebra A to be quaternion if it is generated by x, y with therelations x2−a, y2−b, yx+xy−2c, for some a, b, c in R. We will say A is non-degenerate if a or c is a regular element of R, and b or c is a regular elementof R. We will be considering the case where R is a domain and ab− c2 6= 0since (ab− c2) is the reduced discriminant of A. Let R1 = R[ε]/ε2.

Proposition 6.2 Let R be a commutative k-algebra. Let a, b, c ∈ R so thatthe quaternion algebra A above is non-degenerate.

1. Any first order deformation over R1 of the quaternion algebra A isisomorphic to an R1-algebra A1 generated by x, y satisfying the relationsx2 − a− αε, y2 − b− βε, yx+ xy − 2c− 2γε, where α, β, γ ∈ R.

2. If A1 is given by α, β, γ and A′1 is given by α′, β′, γ′ then A1 ' A′1 iff(α, β, γ) ≡ (α′, β′, γ′) modulo the relations (2a, 0, c), (2c, 0, b), (0, 2c, a), (0, 2b, c).We let

M =

2a 2c 0 00 0 2c 2bc b a c

.

SoAlgExtR(A,A) ' cokM.

Proof. We proceed by using the complex in proposition 3.12 as in thediagonal case. The four overlaps x3, y3, y2x, x2y yield the conditions

[x, α] = 0, [y, β] = 0,

36

[β, x] + 2[γ, y] = 0,

[y, α] + 2[x, γ] = 0.

Writingα = α00 + α01x+ α10y + α11xy,

and similarly for β, γ, we get

0 = [x, α] = 2(α10xy − cα10 + aα11y + cα11).

So by the non-degeneracy hypothesis, α10 = α11 = 0. So we write α =α0 + α1x and similarly β = β0 + β1y. Now the third overlap gives us

0 = [y, α] + 2[x, γ] = 2((cα1 − 2cγ10 + 2cγ11) + 2aγ11y + (2γ10 − α1)xy).

Using the non-degeneracy hypothesis again, we get that γ11 = 0 and 2γ10 =α1. So we may write

γ = γ0 +1

2β1x+

1

2α1y,

completing the description of the kernel.We now determine the image resulting from infinitesimal changes in x

and y. We again denote these changes by d(x) and d(y). We compute thatα is taken modulo

xd(x) + d(x)x = 2((d01(x)a+ d10(x)c) + (d00(x) + d11(x)c)x).

So γ is taken modulo

d(y)x+ yd(x) + xd(y) + d(x)y =

2(d00(y)x+d01(y)a+d10(y)c+d12(y)cx+d00(x)y+d01(x)c+d10(x)b+d11(x)cy).

We can kill α1, β1 by using d00(x), d00(y). We see that α0, β0, γ0 may be cho-sen in R modulo the presentation matrix in the statement of the propositionwhere the relations are generated by d01(x), d10(x), d01(y), d10(y).

We now compute S(A) and θ for the quaternion algebras. Since ID =dR = (ab− c2)R is a principal ideal we may identify NX/D with R/dR. The

37

map AlgExtR(A,A)→ NX/D may be written as

R4 R3 AlgExtR(A,A) 0

R R NX/D 0.

M1001

ba−2c

δ

By the right exactness of cokernel, the sequence

AlgExtR(A,A)→ NX/D → R/(a, b, c)→ 0.

is exact.Let du = ∂d/∂u, dv = ∂d/∂v. Suppose du, dv form a regular sequence and

d ∈ (du, dv). This holds in the cases II ik and IIIk we are considering. Ifd = αdu + βdv then the sequence

0→ Θ(− logD)→ ΘR → NX/D → T1 → 0

may be written as

0 −−−→ R2

(α −dvβ du

)−−−−−−−−→ R2 (du,dv)−−−−→ R/dR −−−→ R/(du, dv) −−−→ 0.

Let

∂(a, b, c)

∂(u, v)=

au avbu bvcu cv

Computations show that the columns of

∂(a, b, c)

∂(u, v)

(α −dvβ du

)are in the column space of M in each of the cases II ik and IIIk. So theinduced map from Θ(− logD) to ker(AlgExtR(A,A)→ NX/D) is zero. This

38

shows that OutDerk(A,A) ' Θ(− logD) by using lemma 5.9 as before. Sousing corollary 4.11 we get that the image of AlgExtk(A,A)→ Θ2

R is IDΘ2R.

The module S(A) is presented by the matrix(∂(a,b,c)∂(u,v)

M).

7 Deformations of Orders over Surfaces

Let X be a smooth projective surface over k. Let A be a maximal OX-order.We will write H i for the sheafification of Hochschild cohomology. We haveagain the basic results in proposition 2.2,

H0k(A,M) = (M)A

H1k(A,M) = OutDerk(A,M)

H2k(A,M) = AlgExt

k(A,M)

H3k(A,A) = Obsk(A,A).

Let D be the discriminant locus of A. Let T1 be sheaf of deformations ofD. So T1 is the cokernel of the map ΘX → NX/D. We define S(A) to be thecokernel of the map ΘX → AlgExt

X(A,A). We write K for the canonical

divisor of X. We now extend theorem 5.11 to the global case.

Corollary 7.1 Let A be a maximal order over a smooth surface X. Thenthere are exact sequences

0→ S(A)→ H0(X,AlgExtk(A,A))→ H0(X,OX(−K −D)).

0→ A/OX → Derk(A,A)→ Θ(− logD).

If A has global dimension two then there is an isomorphism

H0(X,AlgExtk(A,A)) ' S(A)⊕H0(X,OX(−K −D)),

and an exact sequence

0→ A/OX → Derk(A,A)→ ΘX → NX/D → T1 → 0.

39

Proof. Since OX(−K) ' Θ2X , and ID is the ideal sheaf of D, we have

that IDΘ2 ' OX(−K − D). We now show the direct sum decompositionof H0(X,AlgExt

k(A,A)) for orders of global dimension two. Since ID ⊗Θ2

X

is locally free, Extp(ID ⊗ Θ2X , S(A)) = 0 for p > 1. So the local to global

spectral sequence

Extp+qX (ID ⊗Θ2X , S(A))⇐ Hq(X,Extp(ID ⊗Θ2

X , S(A))) = Epq2

collapses to give

Ext1X(ID ⊗Θ2

X , S(A)) = H1(X,Hom(ID ⊗Θ2X , S(A))).

Since S(A) has support in SingD, the above module is zero. So the extension

0→ S(A)→ AlgExtk(A,A)→ ID ⊗Θ2

X → 0

is globally split for orders of global dimension two. The rest follows fromtheorem 5.11.

The following propositions follow from standard patching of deformations.

Proposition 7.2 Let A be a maximal order over X.

1. A first order deformation is determined locally up to isomorphism by asection in H0(X,AlgExt

k(A,A)).

2. There is an obstruction to globalizing a local deformation given by amap H0(X,AlgExt

k(A,A))→ H2(X,Derk(A,A)).

3. If the obstruction vanishes, the possible global deformations with thegiven local isomorphism class is a principal homogeneous space underH1(X,Derk(A,A)).

Proposition 7.3 Let S be an Artin local ring with residue field k. Let S ′ bea quotient of S so that the kernel of the map S ′ → S is dimension one vectorspace over k. Let AS′ be a deformation of A over S ′. We consider extendingAS′ to a deformation over S.

1. There is an obstruction to extending the local deformation given by amap H0(X,AlgExt

k(A,A)) → H0(X,Obsk(A,A)). If this obstruction

vanishes the deformation AS′ can be extended to S locally.

40

2. Suppose that this obstruction vanishes. Then the deformation can beextended locally, but the obstruction to finding compatible local defor-mations is in H1(X,AlgExt

k(A,A)).

3. If this obstruction also vanishes, then the set of possible local extensionsis a principal homogeneous space under H0(X,AlgExt

k(A,A)).

4. The obstruction to finding a global deformation over S is inH2(X,Derk(A,A)). If this obstruction vanishes then the set ofpossible global extensions is a principal homogeneous space underH1(X,Derk(A,A)).

The bracket of a first order deformation A is defined to be the image ofthe class of the deformation under the natural map

H0(X,AlgExtk(A,A))→ H0(X,Θ2

X).

as in 4.10. If a first order deformation of an order A gives rise to a non-zerobracket in H0(X,Θ2

X), we must have that H0(X,OX(−K−D)) 6= 0 by 5.11.More generally, if H0(X,OX(−D − K)) = 0 then H0(X,AlgExt

k(A,A)) '

S(A) by 7.1, and so all local deformations of A are deformations as orders.So all deformations of A over arbitrary Artin algebras are finite over theircentres. So in order to find a deformation not finite over its centre, we cer-tainly require that −K is effective. Denote by kod(X) the Kodaira dimensionof X as in [Beau] VII.1.

Proposition 7.4 If the anticanonical divisor −K is effective, then X iseither

1. birational to a ruled surface,

2. a minimal K3 surface or an abelian surface.

Proof. In order for −K to be effective we either have kod(X) = 0 withωX ' OX , or kod(X) = −∞. These are the above cases by the Enriques-Kodaira classification of surfaces as shown in [Beau] VII.3,VI.20,VIII.2.

Proposition 7.5 If A is a maximal order over an abelian surface or a min-imal K3 surface that has deformations with a non-trivial bracket then A isan Azumaya algebra.

41

Proof. Since in both of these cases OX ' OX(−K), D is the trivial divisor.Hence A must be unramified.

We recall some well known generalities about algebraic surfaces, and inparticular ruled surfaces. We will be using [H] as our main reference for theseresults. The following is in [BPV] II.10.

Proposition 7.6 The arithmetic genus of an effective divisor D is given bythe formula

2pa(D)− 2 = D.(D +K).

In particular, pa(−K) = 1. The next statement follows from [H] V 3.2,V3.3,V 3.6.

Proposition 7.7 Let π : Xp → X be the blowup of X at p. Let K ′ be the

canonical divisor of Xp. We denote by C the proper transform of a divisorC in X. Let E be the exceptional divisor. Let m denote the multiplicity at pof the divisor −K. Then

−K ′ = π∗(−K)− E = −K + (m− 1)E.

The effective −K ′ are obtained by blowing up at p ∈ K. If m = 1 then−K ′ ' −K.

We now introduce the notation of [H] V 2.8.1 for ruled surfaces that willbe used throughout this section.

Notation 7.8 Let X be a minimal ruled surface over a curve C of genus g.Then there is a rank two vector bundle V on C so that X ' P(V ). MoreoverV may be chosen so that H0(C, V ) 6= 0 and if L is a line bundle on C withdegL < 0 then H0(C, V ⊗ L) = 0. The number e = − deg(∧2V ) is inan invariant of X. Let OX(1) be the line bundle on X that is dual to thetautological line bundle that associates a point to the corresponding line in V .There is a section C0 of C so that OX(C0) ' OX(1). We will write the linearequivalence class of a divisor in terms of PicX ' ZC0⊕Pic(C)f where f isa fibre.

The following proposition, [H] V 2.18,2.20, is an application of Nakai’scriterion.

42

Proposition 7.9 Let X be a ruled surface over C. If aC0 + Pf is linearlyequivalent to a irreducible curve 6= C0, f then

1. a > 0 and deg(P ) > ae.

2. If in addition X is a rational ruled surface then a > 0, degP > ae ore > 0, a > 0, degP = ae.

The possible discriminant curves of an order with deformations not finiteover its centre are restricted by the Brauer group. We first analyze thecase of a rational surface. A polygon of n smooth rational curves is a sumC = C0 + · · ·+ Cn−1 of smooth rational curves so that Ci meets each of thecurves Ci−1 and Ci+1 transversely in a single point and does not meet theother curves at all, where the indices are taken modulo n.

Proposition 7.10 Suppose A is a maximal order over a rational surfaceX. If A has deformations with a non-trivial bracket then either D = 0 andA = EndV for some vector bundle V on X, or D = −K and −K is one ofthe following

E smooth elliptic curve,

A1 rational curve with one node,

An a polygon of n smooth rational curves.

We will prove this proposition by classifying all possible curves in thelinear system | −K|, and then using the Brauer group to shorten the list tothe above one.

Lemma 7.11 If X is rational surface with −K effective then −K is one ofthe following

E a smooth elliptic curve,

A1 a rational curve with one node,

A′1 a rational curve with one cusp,

A′2 a sum of two smooth rational curves meeting at a single point withorder two,

43

A′3 a sum of three smooth rational curves meeting transversely at a singlepoint,

An a polygon of n smooth rational curves,

F a non-reduced divisor supported on a tree of smooth rational curves.

Proof. We will show that this list of curves is stable under blowing up apoint by using 7.7. Let m be the multiplicity of −K at p, the point to beblown up. If m = 0 then −K ′ is not effective. If m = 1 then −K ′ ' −K. Ifwe blow up at p with m > 2 then −K ' Ai gives −K ′ ' Ai+1, and −K ' A′igives −K ′ ' A′i+1 for i = 1, 2. Also −K ' A′3 and −K ' F both give−K ′ ' F .

So we only need to verify the statement for minimal rational surfaces.The sequence

0→ OX(K)→ OX → O−K → 0

shows that h0(X,O−K) = 1 and so −K is connected. Using the notation of7.8 we write

X = P(OP1 ⊕OP1(−e))

for some e > 0. Now −K = 2C0 + (2 + e)f . Using the genus formula we seethat pa(−K) = 1. For any subcurve C = aC0 + bf with 0 6 a 6 2, 0 6 b 6e+2 and C 6= 0, C 6= −K, we have pa(C) 6 0. So if −K is integral it must beE,A1 or A′1. If −K is reducible it must be a sum of smooth rational curves.If −K is not reduced then pa(supp(−K)) = 0 since −K is connected. So−K must be supported on a tree of smooth rational curves [BPV] II 11(c).

Suppose e > 2, then C0 is a fixed component of −K by the comparisonsequence

0→ OX(−K − C0)→ OX(−K)→ OC0(−K)→ 0.

since −K.C0 = 2 − e < 0. The only smooth rational subcurves of −K arelinearly equivalent to one of f, C0, C0 + ef, C0 + (e+ 1)f, or C0 + (e+ 2)f by7.9. Computing the intersection numbers of these possible subcurves, we seethat the only possible reduced decompositions of −K by using these curvesare A2, A

′2, A3, A

′3 or A4.

If e = 2 then f, C0, C0+2f, C0+2f, C0+3f, C0+4f and−K are all possibleintegral subcurves. We obtain the same possible cases and the possibility that−K itself is integral.

44

If e = 0 then X = P1×P1 then the possible integral subcurves are linearlyequivalent to one of f, C0, C0 +f, 2C0 +f, C0 +2f, and −K. We again obtainthe same possibilities as when e = 2. The last minimal surface to consideris P2. This case follows from looking at all cubic curves in P2. The possiblecases are E,A1, A

′1, A2, A

′2, A3, A

′3, and F .

Lemma 7.12 Let D be the discriminant locus of a maximal order A over asurface X. Suppose f : P1 → D is a nonconstant map. Then f−1(SingD)contains at least two distinct points.

Proof. There is a canonical exact sequence

0→ BrX → BrK(X)→⊕C

H1(K(C),Q/Z)→⊕p

µ[−1]→ µ[−1]→ 0

(7.13)for a simply connected surface X that is proved in [AM]. The argument of[AM] using the sequence 3.1 and 3.2 of that paper show that this sequenceis a complex for any projective surface. This complex is exact except at⊕

H1(K(C),Q/Z). The cohomology at that term is H3(S, µ). Since a max-imal order is contained in the central simple algebra A ⊗ K(X) we get aclass in BrK(X). This sequence gives constraints on the possible ramifica-tion curves D. The order A determines a cyclic extension of K(C) for eachcomponent C of the ramification curve D. This cyclic extension correspondsto a ramified cover of the normalization of the curve C. The exact sequencerequires that the ramification of the covers of the various curves at a point pmust cancel to yield a element of BrK(X). Any nontrivial cover of a smoothrational curve must ramify at two or more points. So if f : P1 → D thenf−1(SingD) contains at least two distinct points.

Proof. We now prove proposition 7.10. The only possible discriminantcurves are the subcurves of 7.11. Using 7.12 we eliminate the casesA′1, A

′2, A

′3, F , and any of their subcurves and the proper subcurves of An

leaving the desired list. The statement for the unramified case follows fromthe fact that the Brauer group of a rational surface is trivial [DeMF] 1.1. Soif A is unramified then A ' EndV for some locally free sheaf V .

We now consider the case of a surface ruled over a higher genus curve.First we describe the possible curves −K.

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Lemma 7.14 Let π : X → C be a birationally ruled surface over a curve Cof genus g > 2 with −K effective. Then

−K = 2C0 + π∗T

where C0 is a section of C and T is a divisor on C. Also, π∗T = T1 + · · ·+Tnwhere the Ti are fibres of the map π. Each Ti is a tree of smooth rationalcurves with Ti.C0 = 1.

Proof. Suppose X is a minimal model and write X = P(V ) for V a rank twovector bundle as above. Now −K is linearly equivalent to 2C0 + Pf whereP ∈ PicC is of degree 2− 2g + e. Since −K is effective, e > 2. Now by thegenus formula and 7.9, the only irreducible subcurves of −K are C0 and f .Using the comparison sequence twice,

|2C0 + Pf | ' |C0 + Pf | ' |Pf |

since C0.(2C0 + Pf) = 2 − 2g − e < 0, C0.(C0 + Pf) = 2 − 2g < 0. So2C0 is a fixed component of −K. The decomposition of −K into irreduciblesubcurves depends only on P . So −K is as described in the statement of thelemma. The statement is preserved when we blow up at a point p in −K byapplying 7.7.

Lemma 7.15 Let π : X → E be a birationally ruled surface over an ellipticcurve with −K effective. Then either

−K = 2E0 + π∗T

where E0 is section of E and T is a divisor on E. Also, π∗T = T1 + · · ·+ Tnwhere the Ti are fibres of the map π. Each Ti is a tree of smooth rationalcurves with Ti.E0 = 1. or

−K = E0 + E ′0

where E0 and E ′0 are disjoint sections of E.

Proof. Suppose X is a minimal model and write X = P(V ) for a rank twovector bundle as above. Now −K is linearly equivalent to 2E0 + Pf whereP ∈ PicE. Since −K is effective, deg(P ) = e > 0, and if deg(P ) = 0 then Pis the trivial divisor. Now by the genus formula and 7.9, the only possible

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irreducible subcurves of −K are E0, f , and E0 + Pf . Using the comparisonsequence, if e < 0,

|2E0 + Pf | ' |E0 + Pf |,

since E0.(2E0 + Pf) = −e. So E0 is a fixed component of −K if e < 0.The possible decompositions of −K are as two disjoint sections E0 andE ′0 ∈ |E0 + Pf | or as 2E0 and a sum of fibres. If e = 0 then X ' P1 × E.In this case −K is linearly equivalent to 2E0 and the linear system of −K isisomorphic to the pullback of the linear system of quadratics on P1. So −Kis as described in the statement of the lemma. The statement is preservedwhen we blow up at a point p in −K by 7.7.

Using 7.12 and the classification of possible anti-canonical curves in 7.14and 7.15, we see that the only possible ramification loci are

1. X is birational to a surface ruled over an elliptic curve and the discrim-inant locus is either

(a) an elliptic curve that is a section of the base curve

(b) two disjoint sections of the base curve.

2. X is birational to a surfaced ruled over a curve B of genus > 2 and thediscriminant locus is a section of B.

Theorem 7.16 Let A be a maximal order over a surface X. Suppose that Xis birational to a surface ruled over a curve of genus > 1. Suppose also thatX has an effective anticanonical divisor −K. If A has deformations with anon-trivial bracket then one of the two following cases holds:

1. A is unramified and is isomorphic to EndV for some vector bundle Von X.

2. X is birational to an elliptic ruled surface and the discriminant curveis the sum of two disjoint elliptic curves E + E ′ ∈ | −K|.

We first need the following proposition on the Brauer group.

Proposition 7.17 Let X be a smooth surface. Let D be a smooth curve inX. Let U = X −D. There is a canonical exact sequence

0→ Br(X)→ Br(U)→ H1(D,Q/Z)γ→ H3(X,µ)→

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H3(U, µ)→ H2(D,Q/Z)γ→ H4(X,µ)→ H4(U, µ)→ 0

where µ is the group of roots of unity in k and γ is Poincare dual to therestriction maps Hp(X,µ)→ Hp(D,µ) for p=0,1.

Proof. Let j : U → X and i : D → X be the inclusions. Let Gm be the sheafof units of OX and µn be the group of nth roots of unity. Local calculationsin [Artin62] IV (3.4), show that Rkj∗Gm = 0 for k > 0 and Rkj∗µn = 0 fork > 1. There is a canonical exact sequence of sheaves on X

0→ Gm → j∗Gm → i∗Z→ 0,

as shown in [SGAIV] p.542, p.609. So by the Kummer exact sequence andthe snake lemma we get the following diagram of sheaves on X,

µn −−−→ j∗µn −−−→ 0y y yGm −−−→ j∗Gm −−−→ i∗Zyn yn ynGm −−−→ j∗Gm −−−→ i∗Zy y y0 −−−→ R1j∗µn −−−→ i∗Z/nZ.

(7.18)

All rows and columns of this diagram are exact with zeroes on the periphery.Given a bounded complex A∗,

0→ A0 → A1 → · · · → An → 0

we will write the image in the derived category as

(A0 → A1 → · · · → An)

Shifting is denoted by A[k]i = Ai+k. We will define three complexes R, S, Tby taking the middle two rows of 7.18. The diagram 7.18 gives the followingquasi-isomorphisms in the derived category of etale OX-sheaves

(µn) ' (Gmn→ Gm) = R, (7.19)

(i∗Z/nZ)[−1] ' (i∗Zn→ i∗Z) = T. (7.20)

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Since Gm is j∗ acyclic,

(Rj∗)(Gmn→ Gm) ' (j∗Gm

n→ j∗Gm) = S.

By the functoriality of Rj∗ applied to the quasi-isomorphism 7.19 on U

Rj∗(µn) ' (Rj∗)(Gmn→ Gm).

Unless otherwise noted all cohomology groups are taken over X. We have ashort exact sequence of complexes 0 → R → S → T → 0 by the diagram7.18. Let Γ be the global section functor. We denote by H the hyperco-homology functor RΓ. The Leray spectral sequence in the derived categoryprovides the quasi-isomorphism

RΓ(X,Rj∗µn) ' RΓ(U, µn),

[Wb] 10.8.3. The cohomology of the left hand side is the hypercohomologyof the complex Rj∗µn. The cohomology of the right hand side is Hp(U, µn).Hence

Hp(X,S) ' Hp(U, µn).

The hypercohomology of R, T may be computed by the quasi-isomorphisms7.19, and 7.20. So the long exact sequence in hypercohomology coming fromthe short exact sequence of complexes is

→ Hp(µn)→ Hp(U, µn)→ Hp−2(i∗Z/nZ)→ Hp+1(µn)→ .

Each complex A∗ has a spectral sequence abutting to its hypercohomology,as in [Wb] 5.7.10, that is formed by taking sheaf cohomology first and thenthe cohomology of the resulting complex

Hp+q(X,A∗)⇐ Hp(Hq(X,A)∗) = Epq2 . (7.21)

Since each of the complexesR, S, T have only two terms this spectral sequencewill give a long exact sequence. By the functoriality of this spectral sequencewe get the following commutative diagram with exact rows and columns

H2(µn) −−−→ H2(U, µn) −−−→ H1(i∗Z/nZ) −−−→ H3(µn)y y yαn

yβnH2(Gm) −−−→ H2(j∗Gm) −−−→ H2(i∗Z) −−−→ H3(Gm)yn yn yn ynH2(Gm) −−−→ H2(j∗Gm) −−−→ H2(i∗Z) −−−→ H3(Gm).

(7.22)

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The long exact sequences given by the spectral sequence 7.21 are writtenvertically. It is shown in [Groth] p.71 that Hq(Gm) is torsion for q > 2. So inthe direct limit over n the maps αn, βn give isomorphisms since Hp(i∗Q/Z) 'Hp+1(i∗Z) and Hq(µ) ' Hq(Gm) via these maps for p > 1 and q > 3 [Milne]2.22. Since Gm is Rj∗ acyclic Hp(X, j∗Gm) ' Hp(U,Gm). So the middleexact sequence of 7.22 gives us the exact sequence to be shown. The topsequence of the diagram 7.22 is the isomorphic to the long exact sequenceinduced by the Leray spectral sequence

Epq2 = Hp(Rqj∗µn)⇒ Hp+q(U, µn).

So the maps H2−q(i∗Z/nZ)→ H4−q(µn) are Poincare dual to the restrictionmap Hq(µn)→ Hq(i∗µn) for q = 0, 1, 2 by [Milne] 11.5,5.4. We now prove the theorem.Proof. Let X0 be birational to a surface ruled over B, a curve of genus > 1.There is a sequence of maps

X0 → X1 → · · · → Xnπ→ B

where the maps Xi → Xi+1 are blow-ups at points and Xn is a P1-bundleover B with projection π. Using the Leray spectral sequence for each of thesemaps we get that the natural restriction map H1(X,µ)←H1(B, µ) is an iso-morphism. So if D is a section of π then the map H1(D,Q/Z)→ H3(X,µ)is Poincare dual to an isomorphism. Hence if D is a single section of theP1-bundle Sn → B then the order A must be unramified. If D is the sumof two disjoint sections then the map H1(D,µ)→ H3(X,µ) is a sum of twoisomorphisms and so cancellation is possible.

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